Another Mind-Crushing Illusion: Straight or Curved Motion?

From the twisted mind of brusspup comes another brain-hurting illusion. This one is really, really convincing, so tell me: When you look at this video, you’re seeing a circle of eight dots rotating as it spins around inside a bigger circle, right?

No, you’re not. As brusspup shows, each individual white dot is moving in a straight line! The trick here is two-fold: One is that the dots aren’t moving at constant velocity (you can see that in the video at the 0:44 mark), and that combined their motion mimics what we’d see if a smaller circle is rolling around inside a big one.

Try as I may, when I look at this video I can’t make my brain see the dots moving linearly; it looks like a circle rolling. If I focus on one of the dots I can see it moving back and forth along a line, but the others still look like the rim of a circle rolling around. For most illusions there’s a moment when your brain can see what’s going on and the illusion shatters, but not with this one. It’s maddening.

When I was a kid, Spirograph was a very popular “game.” It wasn’t really a game, but a set of clear plastic disks with gear teeth around them (or rings with teeth on the inside). They had holes in them; you’d pin a ring down on a piece of paper, then take another disk, place it inside the ring, put your pencil tip in a hole, and roll the inner disk around inside the outer ring. The results were really lovely and graceful interlocking and overlapping curves. If you’re a lot younger than me and missed this craze, here’s a video that’ll help you picture it:

Man, I miss Spirograph. It was so much fun*! And this dots illusion is related. In Spirograph, when you’d use your pencil to roll around the inner disk, the motion you made was very similar to what you’re seeing in the illusion; it was more of a back-and-forth motion than an around-and-around one. It’s difficult to explain without math, which I find funny; I have a visceral feeling for it because of all those hours I spent playing with a Spirograph when I was little.

If you want the math, then here you go: The shapes made this way (tracing the motion of a point on a circle as it rolls) are called cycloids, and there are a lot of varieties: epicycloids, hypocycloids, and others, depending on how the inner circle is rolled. I once modeled the shell of gas around a supernova as an epicycloidic shell (like a peanut shell), and it reproduced what we saw with Hubble pretty well (even though it formed in a very different way than a cosmic Spirograph!).

Cycloid motion: As a circle rolls on a line, a fixed point on its rim traces out a shape called a cycloid.

Photo from Wolfram Alpha

I’ll note that when a circle rolls along a straight line, if you watch a single point on it you can break the motion up into two dimensions: Horizontal and vertical motion. These motions aren’t constant, but depend on the sine and cosine of the time elapsed. They start off motionless, accelerate to a maximum speed in the middle, then slow back down to zero … and in each direction, the point moves linearly! It’s only when you combine them that you get the cycloid.

That’s how this illusion works. By mimicking this trigonometric motion, your eyes and brain are fooled into thinking the dots are acting together, portraying the rim of a circle. But they aren’t; their motions are related but independent of one another (what’s called “parametric” in mathematics).

A-flippin’-mazing. But also MATH! And SCIENCE!

And another in a long, long series of illusions that shows very well that seeing is not necessarily believing. Our brains are very easily fooled, and that’s very important to remember in life.